TITLE: Diagonalizable matrix $A$ invertible also?
QUESTION [17 upvotes]: If a matrix $A$ is diagonalizable, is $A$ invertible? 
I know that $P^{-1}AP = \text{some diagonal matrix}$ and therefore $P$ is invertible, but not sure of $A$ itself.

REPLY [11 votes]: No. For instance, the zero matrix is diagonalizable, but isn't invertible.